Velocity Underrelaxation in SIMPLE type methods
A (somewhat general) description of the SIMPLE and the SIMPLEC algorithm for pressure correction for incompressible flows for implicit time integration is given by:
0. Use values from previous time step to start outer iteration. 1. Form coefficients and solve momentum equations for intermediate velocity or u*. 2. Solve continuity equation (as the pressure correction equation) for p'. 3. Correct the pressure using p'. 4. Correct the velocity using u'. 5. Use corrected velocities and pressures as improved estimates and repeat from Step 1 until values of p' and u' corrections are "small enough". (note this description has been described by numerous authors; the current description was loosely taken from page 168 of Ferziger's Computational Methods for Fluid Dynamics). NOW MY QUESTION: Where do you apply velocity underrelaxation? Do you apply it in Step 1, where the solution of the momentum equations is done for u*, OR in Step 4, where the velocities are updated before the next outer iteration OR does it even matter? Note that if you underrelax velocity in Step 1 it can be done by incorporating the underrelaxation factor in the matrix coefficients in the discrete momentum equation; otherwise the velocity under relaxation could take place after the velocity correction in Step 4, where the new velocity would be a combination of the corrected velocity and the velocity from the previous outer iteration. Thanks, Matt U. 
Re: Velocity Underrelaxation in SIMPLE type metho
As far as I know, the underrelaxation in SIMPLE is integrated into the matrix calculation, in other words, it is embodied in the coefficiants AIP,AIM,CON, and so on.

Re: Velocity Underrelaxation in SIMPLE type metho
If you apply it at 4 the corrected velocity field will not satisfy continuity (because you have reduced the correction). The pressure correction at 3 is usually reduced relative to the velocity correction.
The relaxation is added at 1 and, usually, included in the coefficients of the pressure correction equation. The word usually is occurring because there are lots of variations in the details of pressure correction schemes. 
Re: Velocity Underrelaxation in SIMPLE type metho
Hi,
(1) The momentum equation can be written as follows with the underrelaxation factor written explicitly at the east face, for example; (Ap/alpa)e Ue=(sigma(AnbUnb))e+be+(1alpa)(Ap/alpa)e Ue,n1(1) where n1 is the previous iteration level and alpa is the underrelaxation factor. Redefine (Ap)e=(Ap/alpa)e (2) then the above equation can be written as follows; (Ap)e Ue=(sigma(AnbUnb))e+be+(1alpa)(Ap)e Ue,n1 (3) Solve above equation to get Ue*. (2) In stage 4, we need velocity correction equations. The above equation can be written as follows, Ue=(Hu)e+(Du)e(PpPE)+other terms, where (Du)e=(dy/Ap)e. (Hu)e=(sigma(AnbUnb)+ScdV)/(Ap)e, Note that the underrelaxationfactor is embodied in (Du)e by equation (2) and ScdV is the source term (be) without the pressure gradient term. The velocity correction equation in SIMPLE can be derived as follows, Ue=Ue* + (Du)e(P'pP'E). Note that Ue* is the velocity obtained by Eq.(3), not the velocity at the previous interation level. I hope this helps, Halim Choi 
Re: Velocity Underrelaxation in SIMPLE type metho
yes, the under/overrelaxation is applied during coefficient generation, before solver

Re: Velocity Underrelaxation in SIMPLE type metho
Good point about the velocity field and the continuity equation. Thanks.

Any difference for relaxation in couple solver?
Do you know if there's any difference to apply underrelaxation in couple solver (velocity & pressure are solved at the same time)? Someone told me before that relaxation has little effect on coupler solver. I don't understand why. Is this really true?
Thanks for your advice. 
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